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Theorem lnolin 22256
Description: Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnoval.1  |-  X  =  ( BaseSet `  U )
lnoval.2  |-  Y  =  ( BaseSet `  W )
lnoval.3  |-  G  =  ( +v `  U
)
lnoval.4  |-  H  =  ( +v `  W
)
lnoval.5  |-  R  =  ( .s OLD `  U
)
lnoval.6  |-  S  =  ( .s OLD `  W
)
lnoval.7  |-  L  =  ( U  LnOp  W
)
Assertion
Ref Expression
lnolin  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( T `  (
( A R B ) G C ) )  =  ( ( A S ( T `
 B ) ) H ( T `  C ) ) )

Proof of Theorem lnolin
Dummy variables  u  t  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnoval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 lnoval.2 . . . . 5  |-  Y  =  ( BaseSet `  W )
3 lnoval.3 . . . . 5  |-  G  =  ( +v `  U
)
4 lnoval.4 . . . . 5  |-  H  =  ( +v `  W
)
5 lnoval.5 . . . . 5  |-  R  =  ( .s OLD `  U
)
6 lnoval.6 . . . . 5  |-  S  =  ( .s OLD `  W
)
7 lnoval.7 . . . . 5  |-  L  =  ( U  LnOp  W
)
81, 2, 3, 4, 5, 6, 7islno 22255 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( T  e.  L  <->  ( T : X --> Y  /\  A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( ( u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `
 t ) ) ) ) )
98biimp3a 1284 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T : X --> Y  /\  A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( (
u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `  t
) ) ) )
109simprd 451 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( ( u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `
 t ) ) )
11 oveq1 6089 . . . . . 6  |-  ( u  =  A  ->  (
u R w )  =  ( A R w ) )
1211oveq1d 6097 . . . . 5  |-  ( u  =  A  ->  (
( u R w ) G t )  =  ( ( A R w ) G t ) )
1312fveq2d 5733 . . . 4  |-  ( u  =  A  ->  ( T `  ( (
u R w ) G t ) )  =  ( T `  ( ( A R w ) G t ) ) )
14 oveq1 6089 . . . . 5  |-  ( u  =  A  ->  (
u S ( T `
 w ) )  =  ( A S ( T `  w
) ) )
1514oveq1d 6097 . . . 4  |-  ( u  =  A  ->  (
( u S ( T `  w ) ) H ( T `
 t ) )  =  ( ( A S ( T `  w ) ) H ( T `  t
) ) )
1613, 15eqeq12d 2451 . . 3  |-  ( u  =  A  ->  (
( T `  (
( u R w ) G t ) )  =  ( ( u S ( T `
 w ) ) H ( T `  t ) )  <->  ( T `  ( ( A R w ) G t ) )  =  ( ( A S ( T `  w ) ) H ( T `
 t ) ) ) )
17 oveq2 6090 . . . . . 6  |-  ( w  =  B  ->  ( A R w )  =  ( A R B ) )
1817oveq1d 6097 . . . . 5  |-  ( w  =  B  ->  (
( A R w ) G t )  =  ( ( A R B ) G t ) )
1918fveq2d 5733 . . . 4  |-  ( w  =  B  ->  ( T `  ( ( A R w ) G t ) )  =  ( T `  (
( A R B ) G t ) ) )
20 fveq2 5729 . . . . . 6  |-  ( w  =  B  ->  ( T `  w )  =  ( T `  B ) )
2120oveq2d 6098 . . . . 5  |-  ( w  =  B  ->  ( A S ( T `  w ) )  =  ( A S ( T `  B ) ) )
2221oveq1d 6097 . . . 4  |-  ( w  =  B  ->  (
( A S ( T `  w ) ) H ( T `
 t ) )  =  ( ( A S ( T `  B ) ) H ( T `  t
) ) )
2319, 22eqeq12d 2451 . . 3  |-  ( w  =  B  ->  (
( T `  (
( A R w ) G t ) )  =  ( ( A S ( T `
 w ) ) H ( T `  t ) )  <->  ( T `  ( ( A R B ) G t ) )  =  ( ( A S ( T `  B ) ) H ( T `
 t ) ) ) )
24 oveq2 6090 . . . . 5  |-  ( t  =  C  ->  (
( A R B ) G t )  =  ( ( A R B ) G C ) )
2524fveq2d 5733 . . . 4  |-  ( t  =  C  ->  ( T `  ( ( A R B ) G t ) )  =  ( T `  (
( A R B ) G C ) ) )
26 fveq2 5729 . . . . 5  |-  ( t  =  C  ->  ( T `  t )  =  ( T `  C ) )
2726oveq2d 6098 . . . 4  |-  ( t  =  C  ->  (
( A S ( T `  B ) ) H ( T `
 t ) )  =  ( ( A S ( T `  B ) ) H ( T `  C
) ) )
2825, 27eqeq12d 2451 . . 3  |-  ( t  =  C  ->  (
( T `  (
( A R B ) G t ) )  =  ( ( A S ( T `
 B ) ) H ( T `  t ) )  <->  ( T `  ( ( A R B ) G C ) )  =  ( ( A S ( T `  B ) ) H ( T `
 C ) ) ) )
2916, 23, 28rspc3v 3062 . 2  |-  ( ( A  e.  CC  /\  B  e.  X  /\  C  e.  X )  ->  ( A. u  e.  CC  A. w  e.  X  A. t  e.  X  ( T `  ( ( u R w ) G t ) )  =  ( ( u S ( T `  w ) ) H ( T `
 t ) )  ->  ( T `  ( ( A R B ) G C ) )  =  ( ( A S ( T `  B ) ) H ( T `
 C ) ) ) )
3010, 29mpan9 457 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  ( A  e.  CC  /\  B  e.  X  /\  C  e.  X ) )  -> 
( T `  (
( A R B ) G C ) )  =  ( ( A S ( T `
 B ) ) H ( T `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   -->wf 5451   ` cfv 5455  (class class class)co 6082   CCcc 8989   NrmCVeccnv 22064   +vcpv 22065   BaseSetcba 22066   .s
OLDcns 22067    LnOp clno 22242
This theorem is referenced by:  lno0  22258  lnocoi  22259  lnoadd  22260  lnosub  22261  lnomul  22262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-map 7021  df-lno 22246
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